Three-phase balanced wye-wye system problem

In summary, we can use the given information about the source voltage, load impedance, and power absorbed by the load to solve for two possible values of line impedance, which are Z_line1 = V_an - (3774 * √3 / (3435)) * (10 + j2) and Z_line2 = V_an + (3774 * √3 / (3435)) * (10 + j2). It is difficult to determine which of these line impedances is more likely to occur in an actual power transmission system without knowing more about the specific system and its conditions.
  • #1
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Homework Statement


In a balanced three-phase wye-wye system, the source is an abc-sequence set of voltages and V_an = 120e^(i30°) V rms. The power absorbed by the load is 3435 W and the load impedance is 10 + j2 Ω. Find the two possible line impedance if the power generated by the source is 3774 W. Which line impedance is more likely to occur in an actual power transmission system?

Z_line = ?
Z_load = 10+j2 ohm
P_load = 3435 W
P_source = 3774 W
V_s = 120e^(i30°) V rms

Homework Equations



V_an = I_aA (10 + j2 + Z_line)

P_s = V_an * I_aA cos θ / sqrt(3)

The Attempt at a Solution



I can't seem to figure out a way to solve for any particular variable... I can find that the power of the line, P_line = 339 W, but that's about as far as I can get. The line impedance appears to be solved by:

Z_line = V_an - I_aA (Z_load) / I_aA
(derived from first relevant eq.)

But the current isn't given, and if I derive current I_aA from second relevant equation, then I don't know cos θ. The fact that there is a cos θ, it does make sense that there would be two valid line impedances.

Any thoughts?
 
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  • #2


Hi there,

I can offer some guidance on how to approach this problem. First, let's start by writing out the equations we have and see if we can manipulate them to get the information we need.

From the first relevant equation, we have:

V_an = I_aA (10 + j2 + Z_line)

From the second relevant equation, we have:

P_s = V_an * I_aA cos θ / √3

Since we know the values for P_s, V_an, and Z_load, we can plug those in and solve for I_aA:

3774 = (120e^(i30°)) * I_aA * cos θ / √3

Solving for I_aA, we get:

I_aA = 3774 * √3 / (120e^(i30°) * cos θ)

Now, we can substitute this value for I_aA into our first equation and solve for Z_line:

Z_line = V_an - I_aA * (10 + j2)

Substituting in the value for I_aA, we get:

Z_line = V_an - (3774 * √3 / (120e^(i30°) * cos θ)) * (10 + j2)

We also know that the power absorbed by the load is 3435 W, so we can use this information to solve for cos θ:

3435 = (120e^(i30°)) * I_aA * cos θ

Solving for cos θ, we get:

cos θ = 3435 / (120e^(i30°) * I_aA)

Now we can substitute this value for cos θ into our equation for Z_line and solve for two possible values of Z_line:

Z_line = V_an - (3774 * √3 / (120e^(i30°) * (3435 / (120e^(i30°) * I_aA)))) * (10 + j2)

Simplifying, we get:

Z_line = V_an - (3774 * √3 / (3435)) * (10 + j2)

So, the two possible values for Z_line are:

Z_line1 = V_an - (3774 * √3 / (3435)) * (10 + j2)

Z_line2 = V_an + (3774 * √
 

FAQ: Three-phase balanced wye-wye system problem

1. What is a three-phase balanced wye-wye system?

A three-phase balanced wye-wye system is an electrical power distribution system that consists of three voltage sources connected in a wye configuration and three loads connected in a wye configuration. This type of system is commonly used in industrial and commercial applications.

2. What are the advantages of a three-phase balanced wye-wye system?

One advantage of a three-phase balanced wye-wye system is that it provides a more stable and efficient distribution of power compared to a single-phase system. It also allows for a higher power capacity and reduces the need for large and expensive equipment.

3. How do you solve a three-phase balanced wye-wye system problem?

To solve a three-phase balanced wye-wye system problem, you must first determine the line and phase voltages using the appropriate equations. Then, you can use Ohm's law and Kirchhoff's laws to calculate the current and power in each phase. Finally, you can use the power triangle to find the total power and power factor of the system.

4. What is the significance of a balanced system in a three-phase wye-wye system?

In a balanced system, the three phases have equal magnitudes and are 120 degrees out of phase with each other. This ensures that the total power in the system is evenly distributed and minimizes the risk of overloading any one phase. It also simplifies the calculations needed to solve the system.

5. What are some common problems that can occur in a three-phase balanced wye-wye system?

Some common problems that can occur in a three-phase balanced wye-wye system include voltage imbalances, phase shifts, and harmonic distortions. These issues can lead to power quality problems and can cause equipment failure if not addressed properly. Regular maintenance and monitoring can help prevent and correct these problems.

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